# Factorion

In number theory, a factorion in a given number base ${\displaystyle b}$ is a natural number that equals the sum of the factorials of its digits.[1][2][3] The name factorion was coined by the author Clifford A. Pickover.[4]

## Definition

Let ${\displaystyle n}$  be a natural number. For a base ${\displaystyle b>1}$ , we define the sum of the factorials of the digits[5][6] of ${\displaystyle n}$ , ${\displaystyle \operatorname {SFD} _{b}:\mathbb {N} \rightarrow \mathbb {N} }$ , to be the following:

${\displaystyle \operatorname {SFD} _{b}(n)=\sum _{i=0}^{k-1}d_{i}!.}$

where ${\displaystyle k=\lfloor \log _{b}n\rfloor +1}$  is the number of digits in the number in base ${\displaystyle b}$ , ${\displaystyle n!}$  is the factorial of ${\displaystyle n}$  and

${\displaystyle d_{i}={\frac {n{\bmod {b^{i+1}}}-n{\bmod {b^{i}}}}{b^{i}}}}$

is the value of the ${\displaystyle i}$ th digit of the number. A natural number ${\displaystyle n}$  is a ${\displaystyle b}$ -factorion if it is a fixed point for ${\displaystyle \operatorname {SFD} _{b}}$ , i.e. if ${\displaystyle \operatorname {SFD} _{b}(n)=n}$ .[7] ${\displaystyle 1}$  and ${\displaystyle 2}$  are fixed points for all bases ${\displaystyle b}$ , and thus are trivial factorions for all ${\displaystyle b}$ , and all other factorions are nontrivial factorions.

For example, the number 145 in base ${\displaystyle b=10}$  is a factorion because ${\displaystyle 145=1!+4!+5!}$ .

For ${\displaystyle b=2}$ , the sum of the factorials of the digits is simply the number of digits ${\displaystyle k}$  in the base 2 representation since ${\displaystyle 0!=1!=1}$ .

A natural number ${\displaystyle n}$  is a sociable factorion if it is a periodic point for ${\displaystyle \operatorname {SFD} _{b}}$ , where ${\displaystyle \operatorname {SFD} _{b}^{k}(n)=n}$  for a positive integer ${\displaystyle k}$ , and forms a cycle of period ${\displaystyle k}$ . A factorion is a sociable factorion with ${\displaystyle k=1}$ , and a amicable factorion is a sociable factorion with ${\displaystyle k=2}$ .[8][9]

All natural numbers ${\displaystyle n}$  are preperiodic points for ${\displaystyle \operatorname {SFD} _{b}}$ , regardless of the base. This is because all natural numbers of base ${\displaystyle b}$  with ${\displaystyle k}$  digits satisfy ${\displaystyle b^{k-1}\leq n\leq (b-1)!(k)}$ . However, when ${\displaystyle k\geq b}$ , then ${\displaystyle b^{k-1}>(b-1)!(k)}$  for ${\displaystyle b>2}$ , so any ${\displaystyle n}$  will satisfy ${\displaystyle n>\operatorname {SFD} _{b}(n)}$  until ${\displaystyle n . There are finitely many natural numbers less than ${\displaystyle b^{b}}$ , so the number is guaranteed to reach a periodic point or a fixed point less than ${\displaystyle b^{b}}$ , making it a preperiodic point. For ${\displaystyle b=2}$ , the number of digits ${\displaystyle k\leq n}$  for any number, once again, making it a preperiodic point. This means also that there are a finite number of factorions and cycles for any given base ${\displaystyle b}$ .

The number of iterations ${\displaystyle i}$  needed for ${\displaystyle \operatorname {SFD} _{b}^{i}(n)}$  to reach a fixed point is the ${\displaystyle \operatorname {SFD} _{b}}$  function's persistence of ${\displaystyle n}$ , and undefined if it never reaches a fixed point.

## Factorions for SFDb

### b = (k − 1)!

Let ${\displaystyle k}$  be a positive integer and the number base ${\displaystyle b=(k-1)!}$ . Then:

• ${\displaystyle n_{1}=kb+1}$  is a factorion for ${\displaystyle \operatorname {SFD} _{b}}$  for all ${\displaystyle k.}$
Proof

Let the digits of ${\displaystyle n_{1}=d_{1}b+d_{0}}$  be ${\displaystyle d_{1}=k}$ , and ${\displaystyle d_{0}=1.}$  Then

${\displaystyle \operatorname {SFD} _{b}(n_{1})=d_{1}!+d_{0}!}$
${\displaystyle =k!+1!}$
${\displaystyle =k(k-1)!+1}$
${\displaystyle =d_{1}b+d_{0}}$
${\displaystyle =n_{1}}$

Thus ${\displaystyle n_{1}}$  is a factorion for ${\displaystyle F_{b}}$  for all ${\displaystyle k}$ .

• ${\displaystyle n_{2}=kb+2}$  is a factorion for ${\displaystyle \operatorname {SFD} _{b}}$  for all ${\displaystyle k}$ .
Proof

Let the digits of ${\displaystyle n_{2}=d_{1}b+d_{0}}$  be ${\displaystyle d_{1}=k}$ , and ${\displaystyle d_{0}=2}$ . Then

${\displaystyle \operatorname {SFD} _{b}(n_{2})=d_{1}!+d_{0}!}$
${\displaystyle =k!+2!}$
${\displaystyle =k(k-1)!+2}$
${\displaystyle =d_{1}b+d_{0}}$
${\displaystyle =n_{2}}$

Thus ${\displaystyle n_{2}}$  is a factorion for ${\displaystyle F_{b}}$  for all ${\displaystyle k}$ .

Factorions
${\displaystyle k}$  ${\displaystyle b}$  ${\displaystyle n_{1}}$  ${\displaystyle n_{2}}$
4 6 41 42
5 24 51 52
6 120 61 62
7 720 71 72

### b = k! − k + 1

Let ${\displaystyle k}$  be a positive integer and the number base ${\displaystyle b=k!-k+1}$ . Then:

• ${\displaystyle n_{1}=b+k}$  is a factorion for ${\displaystyle \operatorname {SFD} _{b}}$  for all ${\displaystyle k}$ .
Proof

Let the digits of ${\displaystyle n_{1}=d_{1}b+d_{0}}$  be ${\displaystyle d_{1}=1}$ , and ${\displaystyle d_{0}=k}$ . Then

${\displaystyle \operatorname {SFD} _{b}(n_{1})=d_{1}!+d_{0}!}$
${\displaystyle =1!+k!}$
${\displaystyle =k!+1-k+k}$
${\displaystyle =1(k!-k+1)+k}$
${\displaystyle =d_{1}b+d_{0}}$
${\displaystyle =n_{1}}$

Thus ${\displaystyle n_{1}}$  is a factorion for ${\displaystyle F_{b}}$  for all ${\displaystyle k}$ .

Factorions
${\displaystyle k}$  ${\displaystyle b}$  ${\displaystyle n_{1}}$
3 4 13
4 21 14
5 116 15
6 715 16

### Table of factorions and cycles of SFDb

All numbers are represented in base ${\displaystyle b}$ .

Base ${\displaystyle b}$  Nontrivial factorion (${\displaystyle n\neq 1}$ , ${\displaystyle n\neq 2}$ )[10] Cycles
2 ${\displaystyle \varnothing }$  ${\displaystyle \varnothing }$
3 ${\displaystyle \varnothing }$  ${\displaystyle \varnothing }$
4 13 3 → 12 → 3
5 144 ${\displaystyle \varnothing }$
6 41, 42 ${\displaystyle \varnothing }$
7 ${\displaystyle \varnothing }$  36 → 2055 → 465 → 2343 → 53 → 240 → 36
8 ${\displaystyle \varnothing }$

3 → 6 → 1320 → 12

175 → 12051 → 175

9 62558
10 145, 40585

871 → 45361 → 871[9]

872 → 45362 → 872[8]